I am having a problem with the brakes on my 1929 U MODEL. When the car is being driven the brakes slowly begin to bind, the Plymouth has to be stopped and a bleeding valve opened and closed to release the brakes. What might be the problem?

There are a number of things that can keep your brake pressure from bleeding off. 

You can have some type of obstruction in your brake lines.  If all four wheels are locking then the obstruction may be from the tee in the middle of the car back to the Master Cylinder.

Also there is a tiny drain hole in the master cylinder that is there to allow pressure to equalize on both sides of the drive piston. 

See diagram item 9



Master cylinder piston return spring stuck not allowing the valve to open and bleed off the pressure.  Remote but possible  See item 20 above.

Have you done any work to the brake system lately?  If so that might be a good place to start.  If not then maybe dirt and sludge have accumulated in the lines.  Might be a good time to blow them out and clean the whole system.

For what it's worth my 2 cents,  Chet...

it was the drain hole that stopped my '37P4.  I had "adjusted" the brake pedal.  jim

I recommend rebuilding the master cylinder.  You can do it yourself.  A new (not NOS) master cylinder kit is fairly inexpensive. (Buy from a brake parts specialty company.) 

I would also replace the brake shoe return springs.  Over time they become weak.  If the spring does not pull the brake shoes in, the front shoe will grab (when going forward.)  I bought some new from Roberts for $7 each.